Are you sure image quality is the biggest factor to decide between them? They're both likely good enough for most applications.

In theory the Sony will have an advantage with the bigger sensor, but at the moment their lens range is very limited and hasn't been seen as that great in quality either. Also, bigger sensor often leads to bigger lenses.

I would have to agree with popo. It's your own choice but I'm not sure image quality is necessarily the best factor by which to choose between them. The Sony has the benefit of a larger sensor so the chances are that its low-light performance and noise management are better but as with most interchangeable lens cameras, the lens you use will show a more apparent difference in low-light performance than the sensor. Like popo said, the problem with Sony is the very limited range of lenses for the NEX series, though there is no shortage of lenses you can adapt though probably at the cost of losing your AF.

As you're quite new to photography it's probably worth bearing in mind that in light of how much you're looking to spend, it's really worth getting your hands on both cameras at a local CE/photography store. With (quite) closely matched cameras, what you take photos with is arguably less important that how often you use it - Camera A may have an arguably superior feature set to Camera B but if you find Camera A uncomfortable to hold or its menu awkward to navigate, you may not use it that often and perhaps even waste its potential.

I'm not saying you should bluntly disregard anybody's views of which is "better" but you should try both out and judge for yourself which is better FOR YOU - one size doesn't fit all.

I'd go for the G3 personally, I don't like the fact that although on the Sony you get the advantage of larger sensors, the lenses that you need to mount onto the body are huge, defeating the fact. If I were to choose another brand, I'd probably go for the Olympus E P3, mainly because of its selection of quality lenses (including many excellent primes), good image quality, fast AF and beautiful styling.

If you want something nice and small, you may want to consider the Pentax Q or Nikon J1 or V1, either one with a small pancake prime lens. The Pentax and Nikon both have smaller sensors (although their image quality is still very comparable), allowing them to be smaller than the other cameras. They may be a bit out of your budget, but you won't be disappointed with either one.

Regarding the "Leica" logo on the Panasonics, it's mainly marketing. I believe that the lenses do use Leica optics, but they're nowhere near the quality of those that you'd find on one of their M series rangefinder bodies. The quality of the Leica Lumix lenses for the Panasonics is very similar to those of other brands, they're not superior or inferior to them.

I have just noticed that the Lumix G3 is lenses based image stablisation instead of camera based. So if I use the lense with stablisation, will the result be different from using the camera's based stablisation feature?

Lens-based and sensor-based IS both have benefits and disadvantages over each other. These are some of the benefits/drawbacks in the context of DSLRs but I'm not sure how much is relevant to EVIL/MFT/CSC cameras

Lens-based+ You see the stabilisation at viewfinder level so you can determine if the image appears to be still more easily
+ The IS doesn't work on a one-size fits all basis so each lens' IS is more finely tuned for that particular lens
- If you want the most up to date IS, you'll have to invest in new lenses
- Lenses with IS tend to command a much higher premium than bodies with IS build around the sensor

Sensor-based+ You can use a 50-year-old lens and IS will still function
+ Only the body needs to be upgraded to stay in touch with improved IS technology
- In theory, the stabilisation won't be as precise as lens-based IS since the sensor-based IS is a general purpose system rather than the tailor-made system in IS lenses

In the real world, however, the differences most people will notice are far more apparent in the wallet than in the photos.

Remember though that concerning ILCs, the camera receives the feed for the electronic viewfinder directly from the sensor, so you'll be able to see the stabilization effect when composing (at least from my understanding of how sensor based IS systems work). Otherwise, I personally think that lens based systems are better. Most lenses made for mirrorless systems do have some sort of IS, and the ones that don't, generally pancake primes, are so wide and fast that they shouldn't need any sort of IS. On my DSLR however, I could definitely see the merits of a sensor based system, like the ability to use IS even with my old Nikon primes. However, regarding mirrorless systems, I definitely think that the advantages of a lens bases system outweigh the disadvantages.

Theoretically yes it is possible but the more likely outcome is that the lens and sensor-based IS systems, when working simultaneously, will work independently from one another, both to each other's detriment. For example, the lens will stabilise the image but motion detected in the body will cause the sensor to move to compensate for the apparent motion of the image, which has already been stabilised by the lens, nullifying the stabilisation the lens had already achieved, resulting in the lens stabilising the image to compensate for the motion in the sensor. In short, putting the two systems together will create a vicious circle.